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•       ^* 


AN   INTRODUCTION  TO 


GRAPHICAL    ALGEBRA 


FOR    THE  USE  OF  HIGH  SCHOOLS 


BY 


FRANCIS  E.    NIPHER,    A.M. 

Professor  of  Physics  and  in  charge  of  Electrical  Engineeririg 
in  Washington  University 


NEW   YORK 

HENRY   HOLT   AND   COMPANY 
1898 


AN   INTRODUCTION  TO 


GRAPHICAL   ALGEBRA 


FOR   THE  USE  OF  HIGH  SCHOOLS 


BY 


FRANCIS  E.    NIPHER,    A.M. 

Professor  of  Physics  and  in  charge  of  Electrical  Engineering 
in  Washington  University 


c  ..  .   .,1 ,  J«,  J 


NEW   YORK 

HENRY   HOLT   AND   COMPANY 
1898 


Copyright,  1898, 

BY 

HENRY  HOLT  &  CO. 


r      •      *     •        •     » 


.        .....•.,.♦..  .         . 


ROBERT    D^UMMOND,    PRINTER,    NEW   YORK. 


PREFACE. 

The  author  has  been  asked  to  present  a  paper  for 
the  consideration  of  the  Trans-Mississippi  Educational 
Convention,  to  be  held  at  Omaha  on  June  28,  29,  and 
30,  1898.  The  subject  assigned  was  the  greater 
efTficiency  of  science  instruction. 

It  did  not  seem  desirable  to  merely  suggest  in  a 
general  way  what  appears  to  the  author  the  most 
desirable  change  in  high-school  instruction  in  order 
to  increase  the  efficiency  of  college  work.  The  aim 
has  been  rather  to  show  how  such  changes  may  be 
accomplished  without  radical  departures  from  present 
methods. 

By  injecting  here  and  there  into  the  ordinary  in- 
struction in  algebra  such  material  as  is  found  in  the 
following  text,  new  meaning  will  be  given  to  the 
operations  involved  in  the  solution  of  equations,  and 
new  interest  in  the  subject  may  be  aroused.  The 
study  of  geometry  by  Euclid's  method  requires  a 
large  amount  of  time  which  for  the  average  student 
might  be  more  usefully  employed.  The  study  of 
algebra  and  geometry  as  wholly  distinct  subjects  hav- 
ing no  relation  to  each  other,  gives  to  the  pupil  a 
false  idea  of  the  intellectual  situation  of  to-day.      The 

iii 

260009 


IV  PREFA  CE. 

scientific  investigator  has,  since  Newton's  time,  very 
largely  abandoned  Euclid's  methods  as  applied  to 
scientific  investigation.  It  does  not  therefore  follow 
that  Euclid's  geometry  should  be  banished  from  our 
schools,  but  it  does  seem  proper  to  consider  whether 
some  of  the  time  given  to  it  might  not  be  more 
usefully  spent  in  elementary  analytical  geometry  or 
graphical  algebra. 

The  devotees  of  geometry  and  mathematics  often 
seem  impatient  that  these  subjects  should  be  studied 
except  for  themselves  alone,  and  for  the  intellectual 
enjoyment  and  mental  development  which  they 
afford.  But  while  this  may  be  a  very  proper  mental 
attitude  for  such  men,  it  does  not  follow  that  we 
should  all  adopt  this  view  of  the  matter.  The  laws  of 
the  physical  universe  are  equally  worthy  of  human 
consideration.  And  these  laws  are  most  impressively 
presented  to  the  mind  in  the  symbolic  language  of 
algebra  or  the  graphical  language  of  geometry. 

We  may  forgive  a  civil  engineer  if  he  confines  his 
admiration  to  the  beauties  of  a  properly  designed 
truss;  nevertheless  we  may  find  new  beauties  in  a 
graceful  building  of  which  the  truss  forms  a  useful 
and  necessary  part. 

The    cross-ruled    paper    needed    for    the  student's 

use  in  graphical  solutions  may  be  obtained  from  any 

dealer  in  draughtsmen's  supplies. 

F.  E.  N. 

St.  Louis,  Mo.,  June,  1898. 


,  }      D     3 

\    J      J        3 
•     ',3       J  3 


AN   INTRODUCTION 


TO 


GRAPHICAL   ALGEBRA. 


When  two  physical  quantities  are  so  related  to 
each  other  that  a  definite  change  impressed  upon  one 
is  always  accompanied  by  a  corresponding  definite 
change  in  the  other,  such  relation  is  called  a  physical 
law. 

The  statement  that  the  weight  of  a  cylindrical  rod 
of  iron  is  proportional  to  its  length  is  a  simple  exam- 
ple of  a  physical  law. 

Every  such  law  may  also  be  stated  in  an  algebraic 
equation. 

It  may  also  be  completely  expressed  in  the  graphi- 
cal language  of  geometry. 

The  examples  that  follow  are  designed  to  show  the 
physical  and  geometrical  significance  which  may  be 
attached  to  the  ordinary  equations  usually  discussed 
in  the  algebra  of  the  preparatory  schools. 

I 


?;  WTR'ODUCTION   TO 

I .  The  weight  of  a  wire  or  rod  is  directly  propor- 
tional to  its  leiigth. 

Let  a  represent  the  weight  per  foot  of  the  wire  or 
rod.      Then  the  weight  in  of  /  feet  will  be 

m  =  al (i) 

This  equation  asserts  that  ni  and  al  are  equal.  If 
the  wire  is  of  uniform  size,  then  it  must  follow  that  if 
one  piece  of  it  is  twice  as  long  as  another  it  must 
have  twice  the  weight,  or  that  m  increases  in  a  direct 
ratio  with  /.  However  the  values  of  /  may  differ  in 
different  pieces  of  the  same  kind  of  wire,  the  weights 
also  so  differ  that  ni  -=-  /  is  always  the  same  and  is 
equal  to  a.      The  equation  may  be  written 


or  ni 


m 

a 

7 

7' 

:  / 

^^ 

a  : 

The  direct  ratio  represented  in  (i)  between  ;;/  and 
/  is  the  most  common  of  all  physical  laws. 

The  value  of  merchandise  is  directly  proportional 
to  the  amount.  The  compensation  due  to  a  workman 
is  proportional  to  the  time-interval  of  his  service. 
The  distance  traversed  by  a  body  moving  at  uniform 
speed  is  proportional  to  the  time.  Within  proper 
limits,  the  yielding  of  a  bridge  is  proportional  to  its 
load,  etc. 


GRAPHICAL   ALGEBRA.  3 

2.    If  squares  of  metal  be  cut  from  a  uniform  sheet, 
the  ivcigJit  is  proportional  to  the  square  of  the  length  of 
the  side. 

Let  b  represent  the  weight  per  square  foot  of  the 
metal  sheet,  and  let  /=  the  length  of  one  side  of  any 
square.      Then  the  weight  vi  of  any  square  is 

m  =  br (2) 

This  equation  teaches  that  if  /^  be  doubled  (that  is 
to  say,  if  the  area  of  the  plate  be  doubled),  the  weight 
will  be  doubled.  If,  however,  /  be  doubled,  the 
weight  will  become  four  times  as  great. 

Let  us  assume  by  way  of  example  that  <;i:  =  8  and 
b  =  2.  The  wire  is  in  that  case  to  be  a  rod  weighing 
8  lbs.  per  foot,  and  the  metal  sheet  is  to  weigh  2  lbs. 
per  square  foot.      Equations  (i)  and  (2)  then  become 

m=^^l, yi)' 

m  =  2/' (2)' 

We  may  now  compute  the  weights  in  of  rods 
of  various  lengths  /,  and  of  squares  of  such  metal 
having  various  lengths  /.  They  are  given  in 
Table  i. 

Let  us  now  assume  that  /  has  the  same  numerical 
value  in  the  two  equations.  Physically  this  would 
mean  that  we  are  to  associate  with  any  rod  /  feet  in 


INTRODUCTION   TO 


TABLE   I. 


/. 

VI  =  Weight  of 

Rods. 

Squares. 

o 

I 

2 

3 
4 

5 
6 

7 

8 

9 

lO 

O 

8 

i6 

24 
32 

40 
48 
56 
64 
72 
80 

0 
2 

8 

18 

32 

50 

72 

98 

128 

162 

200 

length  a  square  having  a  side  of  /  feet.  Then  in 
general  m  would  not  be  the  same  in  the  two  equations. 
If,  for  example,  /=  10,  the  weight  of  the  rod  would 
be  80  lbs.,  while  the  weight  of  the  square  would  be 
200  lbs.,  as  the  table  shows.  A  similar  assumption 
of  equality  in  the  values  vi  would  enable  us  to  elimi- 
nate m,  but  we  must  distinguish  between  the  two 
values  /  thus: 

2r  =  8/'. 

If,  however,  we  also  simultaneously  assume  that 
the  /  of  one  equation  is  the  same  as  the  /  of  the  other, 
we  may  write  the  last  equation 

2/(/—  4)  =  O. 

Here  is  an  equation  in  which  the  product  of  two 
factors  is  zero.      This  will  be  possible  if  either  factor 


GRAPHICAL   ALGEBRA.  5 

is  zero.  That  is  to  say,  the  m  of  one  equation  will 
be  the  same  as  the  m  of  the  other  equation  and  the  / 
of  one  will  be  the  same  as  the  /  of  the  other,  either 
when  2/  =  o,  and  therefore  when  /  =  o,  or  when 
/  —  4  =  o,  or  /  =  4. 

In  other  words,  2/"  will  equal  8/  either  when  /  =  o 
or  when  /  ==  4.  When  /  =  o,  the  ;//  of  each  equation 
is  zero,  and  when  /  =  4,  it  is  32  lbs.,  as  the  table  of 
values  shows. 

In  order  to  represent  geometrically  the  nature  of 
these  two  equations,  involving  these  two  laws,  let  us 
lay  off  along  a  horizontal  direction  the  lengths  /rang- 
ing from  o  to  10  (Fig.  i),  and  forming  an  axis  or  scale 
of  lengths.  At  right  angles  to  this  axis  lay  off  any 
other  scale  for  the  values  ;;/.  In  the  figure  this  scale 
ranges  from  o  to  100.  The  scale  value  of  one  inch 
measured  along  the  axes  may  be  the  same  for  in  and 
/,  or  the  scales  may  be  different,  as  convenience  may 
determine. 

Corresponding  to  various  values  /,  along  the  hori- 
zontal scale,  lay  off  distances  ni  as  given  in  the  table. 
If  the  first  column  ;;/  be  taken,  the  points  so  deter- 
mined will  lie  in  a  straight  line  marked  (i)'  in  the 
figure.  The  vertical  distance  between  the  horizontal 
axis  /  and  the  inclined  line  (i)'  is  proportional  to  its 
distance  /  from  the  intersection  at  the  origin.  These 
vertical  lines  may  be  considered  the  bases  of  triangles 
having   a   common  vertex  at   the   intersection.      The 


INTRODUCTION    TO 


bases   of   similar   triangles   are    proportional    to    their 
altitudes. 

If    the   second    column    of   the    table    be    similarly 
diagrammed    with   the   corresponding  values   of   /,    a 


curve  marked  (2)'  is  obtained  as  shown  in  the  figure. 
In  this  case  the  vertical  distance  from  the  horizontal 


GRAPHICAL   ALGEBRA.  7 

axis  to  the  curve  is  proportional  to   the  square  of  /, 

instead  of  to  the  first  power  as  in  the  other  case. 

Equation  (i),  where  a  is  any  constant,  is  said  to  be 

the  equation  of  a  straight  h'ne.      If  t?  =  8,  as  in  (i)', 

the    equation    is    then    the    equation    of   a    particular 

in 
straight  line,  whose  slope  y-  is  8,  and  which  is  shown 

in  Fig.  I.  Similarly,  equation  (2)'  is  the  equation  of 
the  curve  shown  in  that  figure.  The  points  on  the 
straight  line  are  located  with  reference  to  the  two 
axes  on  which  the  scales  are  marked,  exactly  as  points 
on  a  map  are  located  by  their  latitude  north  or  south 
of  the  equator,  and  their  longitude  east  or  west  of  any 
assumed  meridian.  If  one  were  to  consider  a  small 
region  a  few  miles  in  extent  around  the  point  where 
the  meridian  of  zero  longitude  crosses  the  equator,  we 
might  locate  points  by  measuring  the  distances,  in  feet, 
east  or  west  of  the  meridian  and  the  distances  north 
or  south  of  the  equator.  We  could  by  means  of  a 
surveyor's  chain  and  compass  find  the  point  whose 
longitude  is  east  50  ft.  and  whose  latitude  is  north 
400  ft. 

The  points  whose  latitudes  are  8  times  the  longi- 
tudes would  lie  in  a  straight  line.  If  we  call  y  the 
latitude  and  x  the  longitude,  and  write  the  equition 

y  =  8;r, 

we  are  then  confining  our  attention  to  a  definite  series 


8 


INTRODUCTtON    TO 


of  points  lying  in  a  straight  line,   if  we   neglect  the 
curvature  and  irregularities  of  the  earth's  surface.      If 


y 

10 
D 
8 

1 

1 

/ 

/ 

1 

^ 
^ 

■ 

/ 

/ 

« 

/ 

/ 

6 

5 

i 

7 

't 

/ 

/ 

/ 

5^ 

/ 

/ 

/ 

^^ 

4 

1 

/ 

/ 

-1 1*^ 

^^ 

/ 

/ 

/ 

31; 

-hjy 

3 

1  1 

/ 

/ 

j  j 

J 

f 

/ 

^^ 

2 

\ 

/ 

/ 

^^ 

^ 

^\' 

^^ 

.^^ 

1 1    / 

/ 

/ 

/ 

/ 

^ 

V 

-J^ 

• 

1 

^ 

Fig.   2. 


lOX 


we  write  J)/  =  ^,  the  line  of  points  would  be  a  differ- 
ent one.  The  slope  of  the  line  would  be  only  half 
as  great.      The  smaller  the  constant   factor,  the  less 


GRAPHICAL   ALGEBFA.  9 

steep   will    be    the    line.      Such    lines    are    shown    in 

It  is  evident  that  we  may  in  a  similar  way  represent 
by  a  line  the  relation  between  any  other  two  quanti- 
ties when  one  quantity  is  a  certain  number  of  times 
the  other.  This  is  what  has  been  done  in  Fig-,  i.  In 
that  case  the  weight  in  pounds  of  a  rod  is  8  times  its 
length  in  feet. 

If  the  rod  had  only  half  the  section,  then  in  would 
be  4  times  /,  and  the  line  representing  the  relation 
between  weight  and  length  would  be  only  half  as 
steep.  In  a  similar  way,  if  the  sheet  of  metal  were 
only  one  fourth  as  thick  as  that  represented  by  (2)', 
that  equation  would  become  in  =  \P .  The  curve 
representing  the  relation  between  in  and  /  would  be 
like  that  shown  in  Fig.  i,  but  would  lie  between  that 
curve  and  the  axis  /.  This  curve  may  be  easily  laid 
down  upon  squared  paper,  after  a  table  of  values  of  in 
for  various  values  /  has  been  computed. 

It  is  evident  from  what  has  preceded  that  equa- 
tions (i)'  and  (2)',  when  separately  considered,  repre- 
sent entirely  different  relations  between  a  weight  and 
a  length,  and  that  we  may  assign  any  value  to  /,  in 
either  equation,  and  find  in  by  computation  if  we 
know  the  numerical  values  of  a  and  b. 

The  physical  statement  corresponding  to  this  is 
that  we  may  compute  the  number  of  pounds  in  any 
wire,  or  in  any  square  sheet  of  metal,  if  we  know  in 


lO  INTRODUCTION    TO 

the  one  case  the  weight  per  foot,  or  in  the  other  case 
the  weight  per  square  foot. 

The  diagram  Fig.  i  also  permits  us  by  inspection 
to  determine  these  values  of  m  for  various  values  / 
within  its  limits. 

3.  Let  us  suppose  that  a  merchant  wishes  to  issue 
X  copies  of  a  one-page  circular  letter.  He  finds  that 
a  typewriter  will  charge  him  5  cents  a  copy.  The 
cost  y  of  X  copies  will  then  be 

y=^x.     .      .      .      .      .      .     (3) 

A  printer  offers  to  furnish  them  at  the  following 
rates:  $2.10  for  the  first  hundred  and  10  cents  extra 
for  each  additional  hundred. 

For  the  work  of  printing  alone  and  furnishing 
paper  the  printer  charges,  therefore,  yV  cent  a  copy. 
In  the  first  100  copies,  10  cents  covers  the  cost  of 
printing  and  paper,  and  therefore  the  printer  must 
have  charged  $2  or  200  cents  for  setting  the  type  and 
preparing  to  print.  The  cost  of  x  printed  copies  will 
therefore  be,  in  cents, 

J  =  200  +  ^^x (4) 

We  may  now  compute  in  a  table  the  cost  y,  in 
cents,  of  any  number  x  of  copies. 


GRAPHICAL    ALGEBRA. 


II 


TABLE    2. 


y  —  Cost  c 

f  .X  copies. 

Typewriter. 

Printer. 

o 

0 

200 

10 

50 

201 

20 

100 

202 

30 

150 

203 

40 

200 

204 

50 

250 

205 

100 

500 

210 

1000 

5000 

300 

Let  us  now  find  the  values  of  x  and  y  in  (3)  and  (4) 
by  elimination.      Eliminating/,  we  have 

5,r  =  200  -f  0.\X, 
4.gx  =  200, 
X  —  40.8  -(-• 
Hence  f  =  204  -|-. 


These  values  of  x  and  j'  were  obtained  by  assuming 
that  the  x  and  the  j'  of  one  equation  are  the  same  as 
the  X  and  the  y  of  the  other.  This  value  of  x  is  the 
particular  number  of  copies  that  would  cost  the  same 
by  one  method  of  production  as  by  the  other.  The 
value  of  y  is  the  cost  of  this  number.  For  a  less 
number  of  copies  the  typewriter  method  would  be  the 
cheaper.  When  x  is  zero,  the  equations  assume  that 
the  preparation  for  printing  has  been  made,  but  none 


12 


INTRODUCTION    TO 


are  printed.  The  cost  of  preparation  with  the  type- 
writer method  is  zero,  and  by  the  other  method  the 
cost  of  setting  the  type  is  200  cents. 


CO 


a 


y 

oOO 

/ 

/ 

/ 

/ 

JfOu 

/ 

/} 

/: 

^' 

\ 

K 

V 

300 

4 

\ 

/ 

y 

/ 

/ 



i^OO 

/ 

~F 

¥i 

ill 

er 

/ 

/ 

/ 

100 

/ 

/ 

/ 

/ 

Ni 

cTriber  c 

/ 

copips 

10       iJO       30       40       50       60       70       SO 
Fig.  3. 


00    100 C 


These  two  equations  (3)  and  (4)  are  also  the  equa- 
tions of  lines.      They  are  shown  in  Fig.  3. 

The  values  of  x  and  /  just  found  by  elimination 


GRAPHICAL    ALGEBRA.  13 

determine  the  point  where  these  lines  intersect. 
These  Hnes  show  to  the  eye  that  for  a  small  number 
of  copies  the  typewriter  method  will  be  the  cheaper, 
while  for  a  greater  number  than  40  the  printing 
method  will  be  cheaper.  As  is  shown  in  the  table, 
the  cost  of  1000  copies  will  be  $50  by  one  method 
and  only  $3  by  the  other. 

4.  The  keeper  of  a  notion  stand  finds  that  on 
Jan.  I,  1898,  his  cash  and  stock  exceed  his  debts  by 
$20.  He  is,  however,  losing  each  day  $2.  At  the 
end  of  X  days  after  Jan.   ist  he  will  be  worth 

/  =  20  —  2^- (5) 

On  the  same  date  another  merchant  finds  that  his 
debts  exceed  the  value  of  his  cash  and  stock  by  $15, 
but  he  is  making  a  net  profit  each  day  of  $3.  At  the 
end  of  X  days  he  will  be  worth 

J  =_  1 5  4-  3,1- (6) 

Equation  (5)  represents  the  business  career  of  the 
first  dealer.  He  will  soon  have  lost  all  his  money. 
The  value  of  y  in  that  equation  will  then  be  zero. 
We  can  find  when  this  will  happen  by  making  j  =  o 
in  that  equation.      It  then  becomes 

o  =  20  —  2X, 


14  INTRODUCTION    TO 

This  shows  that  x  =  lO  when  y  =  o.  His  wealth 
will  become  zero  on  Jan.  nth,  or  ten  days  after  Jan. 
1st.  If  he  continue  in  business  he  will  be  going  into 
debt. 

If  we  make  J  =  o  in  equation  (6),  we  have 

o  =  —  15  +  3^, 
or  X  —  S- 

This  shows  that  on  Jan.  6th,  the  second  dealer  will 
be  out  of  debt. 

If  we  find  y  and  x  in  equations  (5)  and  (6),  we  have 

20  —  2^-  =  —  1 5  +  sx, 

5-1' =  35, 
;r  =  7. 

This  shows  that  seven  days  after  Jan  1st  these 
men  will  have  equal  fortunes.  Making  x  =  7  in 
either  (5)  or  (6),  we  find  that  j/  =  -|-  6,  which  will  be 
what  each  will  be  worth  at  that  time. 

By  assigning  consecutive  values  to  x  in  (5)  and  (6) 
we  find  the  fortune  of  these  men  on  successive  days 
after  Jan.   1st  as  shown  in  Table  3. 

The  sinking  and  rising  fortunes  of  these  men  are 
also  shown  in  Fig.  4.  The  line  marked  (5)  in  Fig.  4 
starts  at  a  distance  20  above  the  ;tr-axis,  and  sinks  to 


GRAPHICAL    ALGEBRA. 


15 


that  axis  where  x  ^  \o.  Prior  to  that  time,  the  dis- 
tance from  the  ,t--axis  up  to  the  inclined  line  represents 
his  net  resources  or  his  wealth.  After  that  time,  the 
distance  from  the  A'-axis  down  to  the  inclined  line 
represents  his  net  liabilities.  His  wealth  is  then  a 
negative  quantity.  In  a  similar  way  the  line  marked 
(6)  shows  the  rising  fortunes  of  the  second  dealer. 
The  values  obtained  by  elimination,  or  /  =:  6  when 
^  =  7,  show  when  these  lines  cross  each  other,  and 
the  distance  from  the  axis  x  up  to  the  point  of  inter- 
section represents  what  each  will  then  be  worth. 

TABLE    3. 


y  =  Fortune  of 

Date. 

0 

Dealer  i 

Dealer  2. 

+  20 

-  15 

-Jan.  I 

I 

+  18 

—   12 

"      2 

2 

+  16 

-     9 

"      3 

3 

+  14 

-     6 

"      4 

4 

+    12 

-     3 

"     5 

5 

+    10 

0 

"     6 

6 

+    8 

+     3 

"     7 

7 

+    6 

+     6 

"     8 

8 

+    4 

+    9 

"     9 

9 

+     2 

+  12 

"   10 

10 

0 

+  15 

"   II 

15 

—  10 

+  30 

"   16 

20 

—  20 

+  45 

"  21 

5.    The  area  of  a  field  is  sixteen  square  miles.      Its 
north  and  south  sides  have  a  length  y,  and  its   east 


i6 


INTRODUCTION   TO 


and  west  sides  have  a  length  x.     Then  by  the  condi- 
tion imposed 


xy 


i6. 


(7) 


It  is  evident  that  either  x  ox  y  may  have  any  value 
between  zero  and  an  infinite  value,  but  the  other  must 
then  have  a  definite  value  such  that  their  product  is 
i6. 


y 

■^50 

+  30 
+  20 
-{■10 
0 
-10 

—  20 

—  30 


/ 

y 

W 

/ 

X 

•\ 

X 

y 

^^ 

k; 

^ 

l^ 

^ 

6 

S 

■m^ 

12 

u 

16       . 

S     2L 

y 

(^ 

•^ 

Fig.  4, 


If  we  assume  values  of  x,  we  may  compute  what  y 
must  be  in  order  that  their  product  may  be  16.  For 
convenience    in    making    the    diagram   the   fractional 


GRAPHICAL   ALGEBRA.  1 7 

values  in  the  table  are  also  reduced  to  decimal  form. 
It  is  evident  that  if  the  breadth  of  the  field  were  only 
a  thousandth  of  a  mile,  the  length  must  be  16000 
miles,  and  as  either  7  or  x  approaches  zero,  the  other 
must  approach  an  infinite  value. 


TABLE    4. 


X. 

y- 

16 

=  1.60 

10 

16 

16 

8  =  ^-^'^ 

=   2.23 

0 

16 

—  =  3-33 

16 

—  =  4.00 
4 

16 

3  ~    ^'^^ 
16 

=  8.00 

2 

9.00 
10.00 

10 

9 

8 

7 
6 

5 
4 
3 

2 

16 

—  =  1.60 
10 

The  values  in  Table  4  giving  the  corresponding  sides 
of  the  field  may  be  laid  off  on  the  axes  y  and  x  of 
Fig.  5,  to  the  scale  of  miles  there  shown.  Assuming 
any  length  to  represent  one  side  of  the  field,  the  other 


i8 


INTRODUCTION    TO 


length  is  so  chosen  that  the  product  xy  is  i6.      This 
is  in  effect  equivalent  to  assuming  one  corner  of  tli 
field  to    be   at   the   intersection  of   the   two  axes,  th. 
sides  adjacent   lying  along  the   axes.      The   opposite 
corner  of  the  field  is  at  some  point  on   the  curve  of 


lOX 


Fig.  5.     The  sides  of  the  fields  are  in  an  inverse  ratio. 
If  one  is  doubled  the  other  must  be  halved. 


GRAPHICAL   ALGEBRA.  1 9 

6.  Let  us  assume  that  the  sides  of  a  field  vary  as 
before,  and  that  the  number  of  miles  around  it  is  20. 
Between  what  limits  may  the  lengths  of  the  sides 
vary  ? 

By  the  condition  imposed,  2x  -j-  2y  =  20,  or 

X  -\-  y  :=  10 (8) 

For  computation  of  y  for  assumed  values  of  x  we 
may  write  the  equation 

J'  =  10  —  X. 

It  is  evident  that  \[  x  =  10,  y  will  equal  zero.  If 
X  were  assumed  greater  than  10,  y  would  be  negative. 
In  this  case  negative  values  of  either  x  or  y  do  not 
admit  of  simple  physical  interpretation,  and  they  need 
not  be  here  discussed.  When  x  =  o,  y  will  be  10,  or 
the  two  long  sides  of  the  field  would  alone  require  20 
miles  of  fencing.  If  x  were  less  than  zero,  or  nega- 
tive, y  would  be  greater  than  10,  and  these  two  long 
sides  would  exceed  the  limiting  value  of  20  miles. 
We  shall  then  assume  values  of  x  between  o  and  10, 
and  shall  for  each  case  compute  the  area  of  the  field 
so  determined.      See  Table  5. 

It  is  evident  that  if  equations  (7)  and  (8)  are  sepa- 
rately considered,  we  cannot  say  that  x  or  y  have  any 
particular  values.  We  have  given  various  values  to 
X  and  have  computed  y.      The  values  of  x  and  y  from 


20 


INTRODUCTION    TO 


the  last  table  have  been  plotted  in  Fig.  5  as  in  the 
previous  table.  They  determine  the  inclined  straight 
line. 

TABLE    5. 


X. 

y  =  \o  —  x. 

Area  =  xy. 

0 

10 

0 

I 

9 

9 

2 

8 

16 

3 

7 

21 

4 

6 

24 

5 

5 

25 

6 

4 

24 

7 

3 

21 

8 

2 

16 

9 

I 

9 

10 

0 

0 

Let  us  now  assume  that  the  x  and  the  y  of  equation 
(7)  are  the  same  as  those  of  (8),  and  find  x  and  y  by 
elimination.  The  physical  meaning  of  this  is  that  we 
are  to  find  the  sides  of  a  field  which  must  have  an  area 
of  16  square  miles,  and  it  must  also  require  20  miles 
of  fence  to  enclose  it.  We  have  then  the  two  equa- 
tions 


or 


Hence 


xy 

2x  -\-  2y 

X  '\-  y 

;r(io  —  x) 

X*  —  10^ 


16, 
20, 
10. 

16, 
-  16. 


GRAPHICAL   ALGEBRA.  21 

Adding  25  to  both  members, 

x"  —  io,r  -J-  25  =  9, 

^  -  5  =  ±  3, 
^'  =  8  or  2, 

When  X  —  8,  j  must  equal  2,  and  when  x  =z  2,  y 
must  equal  8.  These  are  the  values  corresponding  to 
the  two  points  of  intersection  of  the  curve  and  the 
straight  line  in  Fig.  5. 

If  2x  -j-  2)'  =  16  instead  of  20,  the  straight  line  of 
Fig.  5  would  drop  two  divisions,  passing  through 
divisions  8  on  the  axes.  It  would  then  be  tangent  to 
the  curve.  The  values  of  x  and  y  determined  by 
elimination  would  then  be  y  =  4,  x  =  4.  This  would 
mean  that  the  field  must  be  square,  with  its  sides 
4  miles  long.  This  is  the  only  possible  case  that  will 
satisfy  the  condition  that  the  area  of  the  field  is  to 
be  16  square  miles  and  the  distance  around  it  is  to  be 
16  miles. 

If  X -\- y  is  assumed  less  than  8,  the  straight  Hne 
will  not  intersect  the  curve.  It  is  impossible  to  have 
a  rectangular  field  of  16  square  miles,  and  so  shape 
the  field  that  less  than  16  miles  of  fence  will  be 
required. 

The  equations  will  tell  the  same  story.  Take 
general  values  for  the  constants  and  write 

xy  —  a, 

b 

X  A-     =  — . 
'  2 


22  INTRODUCTION   TO 

Eliminating  y. 


4-') 


x\—  —  ^j  =  «, 


X X  ==  —  a. 

2 


V76 


-^=-±\/t^-«- 


In  this  we  see  that  if  — >  is  less  than  a,  the  quantity 

lo  ^  ■' 

under  the  radical  will  be  negative    in  sign,   and   the 

extraction    of    its    square    root    is    impossible.      This 

would   mean   that  an   impossible  condition   had    been 

assumed.      In    the    prior   example    we    made  a  =  i6. 

// 
If  — ;  =  rt  =  i6,  then  b  =  i6.      The  value  of  x  would 

ID 

then  be  \b  z=  4. 

In  general  we  find   that  when  the  distance  around 

b'  h 

any    field    of    area    a    is    least,    —^^.a   or   —  =  4^, 

^  ID  4 

Stated  in  words  this  means  that  one  fourth  of  the 
distance  around  the  field  is  equal  to  the  square  root  of 
the  area.  This  can  only  be  true  when  the  field  is 
square  in  form. 

7.    Let  us  have  given  us  the  two  equations 

J  +  ^  =  4, (9) 

J  +  -^-=  2 (10) 


GRAPHICAL    ALGEBRA. 


23 


Viewing  these  equations  from  a  common-sense 
standpoint,  we  should  at  once  decide  that  whatever 
y  and  x  may  be,  their  sum  cannot  be  botli  4  and  2. 
But  it  is  also  true  that  their  sum  may  be  either  4  or  2. 
If  X  =  1000,  then  if  y  be  —  996  the  sum  will  be  4, 
while  if  y  be  —  998  it  will  be  2.  In  other  words,  y 
and  X  cannot  have  simultaneous  values  in  the  two 
equations. 

We  will  asume  various  values  of  x  ranging  from 
—  4  to  -|-  6,  and  compute  the  values  of  y  from  equa- 
tions (9)  and  (10).  These  values  are  given  in  columns 
2  and  3  of  Table  6. 

TABLE   6. 


-r. 

y- 

(9) 

(10) 

-  4 

-  3 

0 

-  I 
0 

+  I 

+  2 

+  3 
+  4 
+  5 
+  6 

+  8 
+  7 
+  6 

+  5 
+  4 
+  3 
+  2 

+  1 
0 

—  I 

-  2 

+  6 
+  5 
+  4 
+  3 

+  2 

+  I 
0 

—  I 

—  2 

-  3 

-  4 

The  values  of  x  and  y  from  (9)  are  laid  off  in  a 
diagram  in  Fig.  6.  They  determine  the  upper 
straight  line.  The  values  of  y  from  (10)  being  laid 
off,  we  get  the  lower  straight  line. 


24 


INTRODUCTION   TO 


+ 

y 

\ 

h 

/ 

\ 

G 

\ 

IJ 

\ 

/^ 

5 

\ 

\ 

4 

\ 

r// 

\ 

\ 

? 

\ 

(0) 

\ 

r/o) 

\ 

—  X 

\ 

\ 

-irX 

4 

3 

1 

1 

\ 

J 

\ 

o          6 
C 

^ 

\ 

\ 

3 

\ 

c' 

4 

\ 

— 

y 

Fig.    6. 


GRAPHICAL   ALGEBRA.  2$ 

These  lines  are  parallel  to  each  other.  There  is 
therefore  no  point  on  the  lines  where,  the  x  of  one 
being  the  same  as  the  x  of  the  other,  the  j/  of  one  will 
be  the  same  as  chej'  of  the  other.  The  two  points 
c,  c'  have  the  same  value  of  x  =  -|-  5>  but  the  corre- 
sponding y  of  one  is  —  i  and  of  the  other  is  —  3. 
The  points  a,  a'  have  a  common  x  ^  —  i,  but  the 
values  J/ are  -|-  3  and  -j-  5.  The  points  a,  b'  have  the 
same  value  j  =  -|-  5,  but  the  values  of  x  are  —  i  and 

-  3- 

This  is  the  geometrical  explanation  of  the  fact  that 

when  we  assume  that  x  and  /  are  alike  in  (9)  and  (10) 
and  seek  to  determine  their  value  by  elimination,  we 
find  ourselves  baffled  by  the  fact  that  we  cannot  elimi- 
nate one  without  eliminating  the  other.  The  equa- 
tions are  both  true,  but  they  are  not  simultaneous. 

If  in  place  of  equation  (10)  we  take  either  of  the 
equations 

,l-+  2J'=  4, (11) 

2.t'+    j=8, (12) 

and  combine  them  with  (9),  we  get  in  either  case  the 
values  J  =  o  and  x  =  -\-  4.  Both  of  the  lines  repre- 
sented by  these  equations  will  cross  the  line  represent- 
ing equation  (9)  at  the  point  where  it  crosses  the 
axis  X.  One  of  them  will  cross  the  axisj'  at  a  point 
where  y  =  8,  and  the  other  where  j  =  2.  This  is 
easily  seen  by  inspection  of  the  two  equations.      K  y 


26  INTRODUCTION    TO 

be  made  equal  to  zero,  both  of  these  equations  give 
for  X  the  value  4.  This  is  also  true  for  equation  (9). 
If  X  =  o,  y  =  4  for  (9),  and  for  (11)  and  (12)  when 
A'  =  o,  J'  =  2  and  8. 

8.    Find  x  and  y  in  the  following  equations: 

.r -f  ^j/ =  84,      .     .     .      .     (13) 

x""  —  y  =  24 (14) 

Solving  these  equations  for  j',  we  have 

7  =  ^--; (13/ 


y=±Vx-'-2A (14)' 

Assuming  values  of  x  between  —  10  and  -f-  10,  we 
have  the  values  of  j'  in  Table  7. 

In  equation  (14)  the  value  of  y  becomes  zero  when 
x^  =z  24  or  X  =  4.899.  The  sign  ±  indicates  that 
for  each  value  of  x''  there  are  two  equal  values  of  / 
with  unlike  sign.  The  axis  x  is  therefore  a  line  of 
symmetry.  When  x"  is  less  than  24,  the  value  o(  y 
becomes  imaginary.  The  value  of  y  is  also  the  same 
whether  x  is  positive  or  negative  in  sign,  since  x  is 
involved  to  the  second  power.  Hence  between 
;tr  =  -f-  4/24  and  —  1^24  the  equation  does  not  give 
real  values.  Each  of  the  curves  obtained  by  laying 
off  the  values  of  y  for  (13)  and  for  (14),  with  the 
values  of  x  given  in  the  table,  consists  of  two  isolated 


GRAPHICAL   ALGEBRA. 


27 


branches  as  is  shown  in  Fig.  7.  They  intersect  at 
two  points  X  —  -\-  'J  and  j'  =  +  5,  and  ;tr  =  —  7  and 
J  =  —  5.  These  are  the  values  that  are  found  by 
making  the  equations  simultaneous  and  solving  by 
elimination. 

TABLE    7. 


Value 

oi  y. 

J 

(■3) 

(14) 

10 

+     1.60 

±   8.72 

— 

9 

-     0  33 

±  7-55 

— 

s 

—      2   50 

±  6.32 

— 

7 

—     5-00 

±  5-00 

— 

6 

—    8.00 

±  3-46 

— 

5 

—  11.80 

±  1. 00 

— 

4 

—  17.00 

— 

3 

—  25.00 

— 

2 

—  40.00 

— 

I 

—  83.00 

0 

00 

+ 

I 

+  83.00 

+ 

2 

-j-  40.00 

+ 

3 

+  25.00 

+ 

4 

-^  17.00 

+ 

5 

+  1 1.  So 

±  1. 00 

-V 

6 

+    8.00 

±  3-46 

+ 

7 

+    5-00 

±  5-00 

+ 

8 

+    2.50 

±  6.32 

+ 

9 

+    0.33 

±  7-55 

+ 

10 

—     1.60 

±  8.72 

9.    Given  the  equation 

40oy  -\-  9^'  —  200xy  -j-  64  =  o. 
This  equation  when  solved  for/  becomes 


(15) 


1/  =  -  +  -  \/ x" 
^      4       5 


(15)' 


28 


INTRODUCTION    TO 


\, 

^ 

^ 

V 

\ 

\, 

^x***^ 

.^-^ 

^ 

y 

s 

N 

e^ 

y 

/ 

^ 

^ 

\ 

N 

to 

y 

y 

V, 

I--:! 

^> 

'^ 

f 

^i 

■i 

>o 

■^ 

to 

^ 

^ 

-^ 

"-I 

to'^ 

^J 

<^:) 

^ 

ic 

to 

^ 

»~» 

1 

"^ 

■^^ 

1    ■ 

is 

^> 

1 

^  5 

1 

^ 

■"3 

to 

^^ 

V. 

/• 

^ 

e^ 

^ 

N 

^-^ 

/ 

y 

^ 

^ 

^ 

\ 

V 

/ 

/ 

~v^ 

•^^ 



N 

/ 

^ 

1 

Fig.  7. 


GRAPHICAL    ALGEBRA.  29 

In  this  form  the  equation  shows  tliat  the  value  j'  is 
made  up  of  two  parts,  viz.,  the  term  \x  and  a  term 
having  a  ±  sign.  Both  terms  increase  with  x.  Each 
value  of  y  will  therefore  have  two  values  for  each 
value  of  X.  The  mean  of  those  two  values  will  be  \x, 
since  one  of  these  values  is  \x  increased  by  the 
±  term,  and  the  other  is  \x  diminished  by  an  equal 
value.  This  at  once  shows  that  the  line  whose  equa- 
tion is 

X 

is  a  line  of  symmetry  for  the  curve  whose  equation  is 
(15)'.  This  line  bisects  the  chords  of  the  curve  drawn 
parallel  to  the  axis  j.  It  is  the  line  marked  a,  b  in 
Fig.  8.  The  two  terms  of  (15)'  are  separately  com- 
puted in  Table  8.  For  values  of  x  less  than  2  the 
lb  term  is  imaginary. 

The  two  values  of  j'  for  each  value  of  x  are  plotted 
in  Fig,  8. 

It  is  evident  that  if  either  equation  (13)  or  (14)  be 
combined  with  equation  (15)  there  will  be  four  values 
of  X  and  four  of  j'.  This  is  apparent  from  an  inspec- 
tion of  the  curves  in  Figs.  7  and   8.      Each  branch  of 

(14)  or  (13)   will   intersect  the    corresponding  branch 

(15)  at  two  points.  Two  values  of  ,r  and  j'  will  there- 
fore be  positive  and  two  will  be  negative. 

10.    From    what    has    preceded    it   will   have    been 


30 


INTRODUCTION    TO 
TABLE    8. 


x. 

y- 

4           5 

+ 

lO 

+  2.50      ±    1.96 

+  4.46     or     +  0.54 

+ 

9 

+  2.25       ±    1.76 

+  4-OI      ' 

'     +  0.49 

+ 

8 

-(-  2.00      ±    1.55 

+  3-55      ' 

*     +  0.45 

+ 

7 

+  1-75     ±   1-34 

+  3-09      ' 

'     +  0.41 

+ 

6 

+  1.50     ±  1. 13 

+  2.63      ' 

*     +0.37 

+ 

5 

+  1.25     ±  o.g2 

+  2.17      ' 

'     +  0.33 

+ 

4 

-|-  i.oo     ±0.69 

+  1.69      ' 

'     +0.31 

+ 

3 

-(-  0.75    ±  0.45 

+   I.20        ' 

'     +0.30 

+ 

2.5 

+  0.625  ±  0.30 

+  0.925     ' 

'     +  0.325 

+ 

2 

+  0.50     ±  0.00 

+  0.50        ' 

'     -\  0.50 

+ 

I 
O 

— 

I 

— 

2 

—  0.50    ±  0.00 

—   0.50        ' 

—  0.50 

— 

2.5 

—  0.625  ±  0  30 

-   0.925     ' 

'      -  0.325 

— 

3 

-  0.75     ±  0  45 

—    1.20        ' 

—  0.30 

— 

4 

—  1.00     ±  0.69 

—    1.69 

—  0.31 

— 

5 

—  1.25     ±  0.92 

-2.17        ' 

'     -  0.33 

— 

6 

-  1.50     ±  1. 13 

—    2.63 

'     —  0.37 

— 

7 

-  1.75     +   1.34 

—    3.09        ' 

'     -  0.41 

— 

S 

—  2.00     ±  1.55 

-    3.55        ' 

'     -  0.45 

— 

9 

—  2.25     ±   1.76 

-    4.01         ' 

'     -  0.49 

lO 

—  2.50     ±  1.96 

-   4.46        ' 

'     -0.54 

observed  Lhat  an  equation  of  the  first  degree  between 
two  variables  x  and  y  may  be  represented  by  a 
straight  line.  To  draw  this  line  only  two  points  need 
be  determined.  The  most  convenient  points  aie  the 
intersections  of  the  line  with  the  axes  of  y  and  x. 
Thus  in  the  equation 


X  +  2y 


(16) 


if  y  be  made  zero  we  have  x  =  4.  This  line  crosses 
the  .r-  a.xis  at  a  point  when  x  =  4.  Making  x  =  o, 
we  have  y  =  +  2.      This   line  therefore   crosses   the 


GRAPHICAL   ALGEBRA. 


31 


\ 

A 

1 

\ 

\ 

"0 

\ 

Co 

\ 

\ 

\ 

^i 

\ 

\ 

Ci 

\ 

\\ 

t1 

\ 

\ 

s.-^ 

\ 

\ 

te, 

\ 

ti, 

• 

1 

\ 

K« 

+ 

•^ 

'^^l 

-^ 

Ca 

iNi 

K,  >::> 

K» 

t^ 

Co 

-fe- 

-■'^ 

h-. 

tfc 

\ 

t^ 

\ 

*- 

\^ 

\ 

•il 

\ 

\ 

01 

\ 

\ 

V 

VJ 

\ 

\ 

Cc 

\ 

\ 

s 

c 

r\ 

\ 

Fig.  8. 


32  INTRODUCTION   TO 

axis/,  where  )'  =  -)-  2.  This  line  is  shown  in  Fig.  9, 
where  it  is  marked  (16).      In  the  equation 

;r-  2/  =  4 (17) 

the  intersection  on  the  axis  of  y  is  —  2  instead  of 
-\-  2.      In  the  equation 

-2x^y  =  A, (18) 

the  line  crosses  the  axes  y  and  x  at  points  y  =  -\-  4 
and  ;r  =  —  2.      In  the  equation 

—  2x—y  =  4 (19) 

the  intersections  are  y  =  —  4  and  x  =  —  2.  All  of 
these  lines  are  shown  in  Fig.  9. 

When  the  equation  is  of  a  degree  higher  than  the 
first,  the  line  representing  it  is  in  general  curved. 
The  intersections  with  the  axes,  if  such  intersections 
exist,  may  be  found  in  the  same  way  as  in  the  straight 
line,  but  the  position  of  other  points  of  the  line  must 
be  computed.  At  phices  where  such  lines  bend 
sharply  the  points  so  determined  must  be  closer 
together  than  at  points  where  there  is  little  curvature. 

II.  When  an  equation  contains  three  unknown  or 
variable  quantities,  x,  y,  and  ::,  three  axes  of  reference 
are  used  in  order  to  represent  the  relation.  Such  a 
method  is  in  general  use  in  such  a  case  as  the  location 
of  the  summit  of  a  mountain.     We  should  give  the 


GRAPHICAL   ALGEBRA. 


33 


latitude  and  longitude  of  liie  place,  and  its  altitude 
above  any  assumed  datum  plane,  such  as  the  level  of 
the  sea. 


+ 

V 

\\ 

I 

\ 

) 

3 

^ 

L 

-9 

^ 

U  i 

t  *i  ) 

1 

fl 

»V 

\ 

/ 

[(3;/- 

^ 

\ 

^ 

X 

5 

-^ 

7 

\  (  J 

1 

0 

1 

1 

2 

.^^ 

^ 

0 

-X 

/ 

\ 

2^^ 

fz- 

') 

/ 

^ 

\ 

3 

A 

\ 

4 

Vj 

\ 

y 

Fig.   g. 

Fig.  lO  shows  a  system  of  three  reference  planes, 
intersecting  each  other  at  right  angles,  in  three  axes. 
The  three  axes  intersect  in  a  common  point  called  the 
origin.  Measured  from  the  origin,  a  distance  to  the 
right  along  the  axis  x  is  -\-  x,  and  to  the  left  is  —  x. 
Distances  upward  and  towards  the  observer  are 
respectively  -j- ;:  and  -\- y,  while  distances  in  the 
opposite  directions  have  the  —  sign. 


34 


INTRODUCTION'   TO 


Fig.  II  shows  one  of  the  eight  trihedral  angles  of 
Fig.  lo.  It  is  intersected  by  a  plane  whose  intersec- 
tions with  the  three  reference   planes   make  angles  of 


Fig.  io. 
45°  with   the  three  axes.      The   axes  are   each   .nter- 
sected  at  a  distance  8  from  the  origin.      The  intersec- 
tion  of  the   inclined    plane  with   the   reference  plane 
X,  3  has  for  its  equation 


GRAPHICAL  ALGEBRA. 


35 


This  line  extends  infinitely  in  both   directions,  but 
only  the  part  between   the  axes  z  and  x  is  shown  in 


Fig.  II. 


the  figure.  In  front  of  this  line  a  distance  j'  =  i  from 
the  back  plane  of  x,  ^  is  a  line  in  the  inclined  plane 
whose  intersections  uith  the  bottom  and  left  reference 


3^  INTRODUCTION   TO 

planes  are  distant   8  —  i    or  8  —  j'  from  the  axis  y. 
The  equation  of  this  line  is  evidently 

x+  "  =  8  -  I  =  8  -y. 

Similarly  the  next  line  in  the  figure,  at  a  distance 
y  —  2  from  the  back  plane,  has  the  equation 

;i'  +  ^  =  8—   2=8—  J. 

From  this  it  is  apparent  that  the  equation  of  any 
line  of  this  series  distant  y  from  the  back  plane  is 

.r  -[-  -^  =  8  —  y- 

Evidently  this  applies  equally  to  any  line  inter- 
mediate between  those  drawn  in  the  figure.  And  the 
equation  therefore  represents  the  location  of  any 
point  on  any  such  line,  or,  in  other  words,  any  point 
on  the  plane.      The  equation  may  be  written 

x-\-y-^z^Z (20) 

If  one  starts  from  the  origin  and  travels  along  the 
axis  X  a  distance  2.  and  then  travels  parallel  to  the 
a^:is  J'  a  distance  6,  the  distance  upward  to  the  plane 
will  be  zero.  If  the  last  distance  parallel  to  y  is  5, 
the  distance  up  to  the  plane  w^ill  be  i.  If  the  distance 
along  J/  be  made  7,  the  distance  to  the  plane  parallel 
to  the  axis  z  will  be  —  i.      In  all  cases,  whatever  may 


GRAPHICAL   ALGEBRA. 


37 


be  the  lengths  of  the  paths  along  x,  j,  and  ,cr,  if  the 
sum  of  these  distances  is  -\-  8,  the  journey  will 
terminate  at  a  point  on  the  inclined  plane.  For  this 
reason  equation  (20)  is  said  to  be  the  equation  of  the 
plane. 


Fig.  12 


Fig.  12  shows  the  same  reference  planes,  with  two 
planes  intersecting  them.  The  equations  of  these  two 
planes  are 


2^  +    3J  +    5^  =  30, 


.     .     (21) 


I5;tr  +  20y  +   \2z  =    I20. 


(22) 


38  INTRODUCTION   TO 

That  these  are  the  equations  of  these  two  planes 
may  be  verified  as  follows:  The  equations  are  to  be 
true  for  all  values  of  x^  y,  and  z.  They  will  therefore 
be  true  when  y  =  o.     The  two  equations  then  become 

2;r4-     5^  =  30,        ....      (23) 
\z^x  A^  \2z  =  \20 (24) 

These  are  the  equations  of  the  two  intersecting  lines 
on  the  back  plane  of  .r,  z.  These  equations  hold  for 
each  point  on  those  lines.  They  hold,  therefore, 
when  ;r  =  o.  In  that  case  we  find  from  the  first 
equation  5^  =  30,  or  ;;;  =  6.  This  determines  the 
point  where  this  plane  cuts  the  axis  z.  For  the  other 
plane,   \2z  =  120,  ov  z  =  10. 

In  a  similar  way  if  ^  =  o,  in  equations  (23)  and  (24), 
we  have  for  the  intersections  on  the  axis  of  x, 
2x  =  30  or  X  ^  15,  and  15^-  =  120  or  x  =  8. 

Likewise  if  x  and  z  are  made  zero  in  (21)  and  (22) 
the  intersections  on  y  are  determined  by  the  resulting 
equations.  We  have  3/  =  30  or  /  =  10,  and  20y  — 
120  or  J'  =  6. 

By  elimination  of  x  or  y  in  (23)  and  (24)  we  may 
find  values  for  those  quantities.  They  are  z  =  ^\^-  = 
4. 12  and  X  —  "Vr  =  4-7 1-  ^^^  these  values  of  x  and 
z,  which  determine  the  point  /;  of  Fig.  12,  were 
obtained  on  the  assumption  that  y  —  o.  By  giving 
y  other  values,  other  points  on  the  line  a,  d,  or  a,  b 


GRAPHICAL   ALGEBRA.  39 

produced,    may   be  obtained.      For  example,  if  x  be 
made  zero  in  (21)  and  (22),  we  have 

3j'  +  5-=30,      .      .      .      .      (21)' 
5J  +  3^  =  30 (22)' 

By  elimination  we  find  j'  =  ^  ^  J^^  =  3.75.  In 
fact,  if  we  assume  x,  y,  and  z  to  mean  the  same  in 
(21)  and  (22),  which  are  the  equations  of  the  two 
planes,  then  the  values  x,  j',  and  z  can  only  refer 
to  the  points  which  are  common  to  the  two  planes, 
and  which  lie  in  their  line  of  intersection  a,  b.  This 
is  the  geometrical  meaning  of  the  fact  that  the  three 
unknown  quantities  cannot  be  determined  with  only 
two  equations. 

If  we  combine  (21)  and  (22)  by  the  elimination  of 
Xj  we  have 

5J'+5i-=2io (25) 

This  is  the  equation  of  a  straight  line  on  the  plane 
r,  s-  It  is  the  relation  between  the  distances  j'  and  z 
of  points  along  the  line  a,  b.  If  the  points  of  this 
line  be  projected  parallel  to  x  upon  the  line  a,  b" , 
equation  (25)  will  be  the  equation  of  that  projection. 
For  each  point  in  a,  b  will  have  the  same  y  and  z  that 
its  projection  has. 

This  projection  will  therefore  intersect  the  axis  z 
(jl'  being  then  zero  in  (25)),  where  z  :=  ^^--  This  is 
the   value   previously  found   for  /;,  b' .      Making  ^  ==  O 


40  INTRODUCTION    TO 

in  (25)  we  find  that  this  projection  intersects  the  axis 
J,  where  J  =  42.      If  y  be  made  ^  in  (25),  we  find  z 
to  be  Jj5.  also,  which  was  previously  determined. 
Eliminating  z  in  (21)  and  (22),  we  have 

51^  +  64J  =  240 (26) 

This  is  the  equation  of  the  line  a  ,  h' ,  which  is  the 
projection  of  a,  b  on  the  plane  x,  y.  Making  j/  =  o 
in  this  equation,  the  distance  to  the  intersection  with 
the  axis  x  at  b'  is  x  =:  -^^Y,  ^^  ^^'^^  previously  found. 

Now  it  is  evident  that  if  we  pass  a  third  plane 
through  this  trihedral  angle  not  parallel  to  either  of 
the  two  planes  of  Fig.  12,  it  will  intersect  each  of 
them  in  two  additional  lines  like  a,  b.  These  three 
lines  of  intersection  will  intersect  each  other  in  a 
common  point,  as  the  intersections  of  the  reference 
planes,  or  the  axes  x,  y,  and  z,  intersect  in  a  common 
point,  the  origin.  This  point  will  be  the  only  point 
which  will  lie  in  all  three  of  the  planes.  That  is  the 
only  point  for  which  x,  y,  and  z  can  have  the  same 
value  in  the  three  equations.  The  values  of  x,  y,  and 
z  obtained  by  elimination  will  locate  this  point  of 
intersection. 

12.  Let  us  assume  the  equations  of  three  planes  as 
follows: 

2x  +  4y  -f  4^  =  20,       .      .      .      (27) 

Sx  +  27  +  5^  =  20,       .      .      .      (28) 
4^  +  SJ'  +  2-s-  =  20.       ...      (29) 


GRAPHICAL    ALGEBRA.  4^ 

There  is  a  line  of  intersection  between  planes  (27) 
and  (28),  another  between  planes  (27)  and  (29),  and 
a  third  between  planes  (28)  and  (29),  Combining 
these  equations  in  that  order,  if  we  eliminate  j  we 
shall  obtain  the  equation  of  the  projections  of  those 
three  lines  on  the  plane  ^,  x.      In  this  way  we  have 


4x  +     3^  =  10,  .      . 

•        •        (30) 

3a- +    6z  =  10,  .      . 

•        •        (31) 

"JX  -\-  2lZ  ~  60.    . 

•        ■        (32) 

Since  the  lines  in  which  the  planes  intersect  each 
other  have  a  common  point  of  intersection,  their  pro- 
jections, represented  by  the  last  equations,  must  have 
a  common  point.  Therefore  any  two  of  those  equa- 
tions will  determine  the  value  of  x  and  ^  for  that 
point.  By  elimination  with  either  (30)  and  (31),  (30) 
and  (32),  or  (31)  and  (32),  we  find  x  =  |o,  ^  =  ||. 

If  these  values  of  .r  and  ::  be  substituted  in  the 
equations  of  either  one  of  the  three  planes,  (27),  (28), 
or  (29),  they  will  each  give  the  value  j  =  |A.  This 
shows  that  the  point  whose  co-ordinates  are  x  =  |-|^, 
J'  =  |P-,  5'  =  H  is  a  point  common  to  the  three  planes. 

The  three  planes  with  their  intersections  are  shown 
in  Fig.   13. 

If  the  coefficient  of  x  in  (27)  were  —  2  instead  of 
-}-  2,  the  intersection  of  that  plane  with  the  ,v-axis 
would  be  at  a  point  —  10  instead  of  -f~  ^O-  The 
points   of  intersection  of   that  plane  with    the   axes  2 


42  INTRODUCTION    TO 

and  J  would  be  unchanged.  The  triangle  determined 
by  the  intersections  with  the  three  axes  would  be  in  a 
symmetrically  reversed  position  in   the  trihedral  angle 


Fig.  13. 


to  the  left  of  the  one  shown  in  Fig.  13.  (See  Fig. 
10.)  This  would  change  the  position  of  two  of  the 
lines  of  intersection  between  the  three  planes,  but 
would  not  change  the  other. 


GRAPHICAL   ALGEBRA.  43 

13,  Instead  of  eliminating  j/  in  equations  (27),  (28), 
and  (29),  as  in  the  last  article,  we  may  reach  precisely 
the  same  result  by  eliminating  either  x  or  z.  In 
either  case  there  will  result  three  equations  like  (30), 
(31),  and  (32),  involving  the  other  two  quantities,  any 
two  of  which  will  determine  those  two  quantities. 
These  quantities  may  then  be  put  into  any  one  of  the 
original  equations,  and  the  values  x  =  ||,  y  =  ||, 
^  =  7|.  will  be  determined.  This  solution  is  merely 
a  projection  of  the  lines  of  intersection  on  the  other 
two  reference  planes. 

If  the  signs  of  the  coefificients  in  the  original  equa- 
tions are  varied,  the  point  common  to  the  three  planes 
may  be  thrown  into  any  one  of  the  eight  trihedral 
ancrles  shown  in  FitJ.  10.  If  x  in  the  final  solution  is 
-|-,  the  common  point  will  lie  in  one  of  the  four  tii- 
hedral  angles  to  the  right  of  the  ]:)lane  y,  z  of  Fig. 
10.  If  )'  is  also  positive,  it  will  be  in  one  of  the  two 
to  the  right  of  y,  z,  and  in  front  of  x,  z.  If  z  is  also 
negative,  it  will  lie  in  the  lower  angle  of  these  two. 

14.  An  equation  of  the  second  degree  between  three 
variables  is  represented  by  a  curved  surface. 

Assume  the  equation 

ZX  +  2J  =  O (33) 

This  surface  may  be  represented  by  lines  represent- 
ing sections  of  the  surface,  as  was  done  in  case  of  the 


44 


INTEODUCTIOX    TO 


plane  in  Fig.  ii.      For  example,  make  x  ^  \o  in  the 
last  equation.      It  then  becoxnes,  on  solving  for  z, 


1    V 


This  is  the   equation  of  a  straight  line.      In  Fig.   14 
this  line  is  the  extreme  line  to  the  right  of  the  model, 


Fig.  14. 

where  x  =  10.      From  the  front  edge  of  the  model  to 
the  rear  section  j  varies  from  o  to  —  10.     The  values 


GRAPHICAL   ALGEBRA.  45 

of  ;:•  in  this  model  are  all  pOGitive,  the  value  at  the 
rear  right-hand  corner,  where  x  =  lO  and  y  =  —  lO, 
being  +2. 

The  straight  line  adjoining   the   previous   one,  and 
along  which  x  =  9,  is  represented  by  the  equation 

z=  -  \y. 

The  line  next  to  the  left  has  for  its  equation 

-  =  —  f J  =  -  \y- 

The   final   line   of   this  series   shown  in   the   figure, 
where  x  =.  i,  has  for  its  equation 

^  =  —  2y, 

Each  of  these  lines  represents  an  intersection  Avith 
the  surface  whose  equation  is  (33)  by  a  vertical  plane 
at  right  angles  to  the  axis  x.  For  all  points  of  this 
plane  x  is  the  same.  Making  x  constant  and  assign- 
ingr  various  constant  values  to  it  enables  us  to  com- 
pute  the  curves  representing  the  sections  of  the 
surface  at   any   distance   x  from    the   reference   plane 

The  other  series  of  sections  shown  in  Fig.  14  as 
crossing  lines  of  the  former  series  are  characterized 
by  the  condition  z  —  constant. 

For  the  uppermost  of  these,  where  ^  =  10, 


46  INTRODUCTION    TO 

For  the  lowermost  one,  where  ^  =  i, 

;f  =  —  2j. 

These  h'nes  correspond  to  intersections  with  the 
surface  made  by  a  horizontal  plane  cutting  the  ver- 
tical axis  at  any  distance  z  from  the  origin. 

The  curved  line  at  the  back  of  the  model,  where 
y  =z  —  lo,  has  for  its  equation 

^;ir  =  -)-  20. 

For  the  upper  point  of  this  curve,  where  the  model 
is  cut  off  on  a  horizontal  section,  ;i-  =  2  and  z  —  10, 
while  for  the  lowest  point  shown  x  =z  10  and  s-  =  2. 
In  both  cases,  and  for  any  other  point  on  this  curve, 
zx  =  20. 

Along  a  similar  section  midway  between  the  front 
and  rear  of  the  model  /  =  —  5.  The  equation  of 
such  section  would  be 

2X  =    10. 

Along  the  front  edge  of  the  model  j'  =  o,  and 
along  this  plane,  as  (33)  shows,  the  product  zx  =  o. 

All  of  these  equations  are  special  cases  of  equation 
(33).  At  any  point  on  the  horizontal  reference  plane, 
determined  by  x  and  y,  the  vertical  distance  z  to  the 
surface  may  be  found  from  (33).      This  value  is 

r 

.^' 

Z  =    —  2  ~. 

X 


GRAPHICAL   ALGEBRA.  47 

This  surface  extends  into  other  angles  formed  by 
the  reference  planes  of  Fig.  lO,  but  it  is  not  thought 
desirable  to  go  further  into  the  subject  here. 

15.  There  are  many  surfaces  in  which  all  of  the 
sections  at  right  angles  to  the  axes  are  curved.  In 
Fig.  14  those  sections  at  right  angles  to  the  axes  z 
and  X  are  not  curved. 

Assume  the  equation 

^xz  —  lyz  -f  6xy  =  0.       ...      (34) 
If  we  solve  this  equation  for  s,  we  have 


.=  -    -^"-^ (35) 


We  will  examine  this  surface  in  the  reference  angle 
where  x  is  positive  and  j  negative  in  sign,  as  in  the 
case  of  Fig.   14. 

If  in  this  equation  we  make  x  —■  10,  (35)  or  (34) 
will  show  the  relation  between  z  and  y,  along  a  section 
of  the  surface,  at  a  distance  x  —  10  from  the  plane 
z,  y.     The  equation  then  becomes 


6or 

2  =^  — . 

40—  ly 


48 


INTRODUCTION    TO 


Giving  y  consecutive  values  between  o  and  —  lo, 
the   values  of  z  are  computed    as  in  table  9. 

Table  9  adjoining.  It  is  evident  r 
that  z  increases  more  slowly  as  y 
becomes  numerically  large.  If  y  = 
—  1 0000,  the  value  of  z  is  only 
19.97.  As  J'  becomes  very  large  the 
value  of  40  in  the  denominator  be- 
comes insignificant  compared  with  3/. 
We  may  under  these  circumstances 
neglect  the  term  40.  As  y  increases 
more  and  more,  z  approaches  the  value 


y- 

2. 

0 

0.00 

—     I 

+  I  39 

—    2 

+  2,61 

~     3 

1-3.67 

-     4 

+  4.61 

-     5 

+  5  45 

-     6 

-t-  6  21 

-     7 

+  6.89 

-     8 

+  7-50 

-     9 

+  8.06 

—  10 

+  8,57 

z  =^  — 


60/ 

-  ly 


+  20. 


This  would  be  the  value  of  z  when  y  is  negative  and 
infinite. 

The  values  of  y  and  z  of  the  table  determine  the 
section  shown  on  the  right  of  Fig.  15,  where  x  = 
+  10. 

\i  X  =^  5,  equation  (35)  becomes 


30J 


20 


2>y 


If  consecutive  values  from  o  to  —  10  be  substituted 
in  this  equation,  we  may  construct  another  table  like 


GRAPHICAL   ALGEBRA. 


49 


the  previous  one.  The  two  tables  with  others  corre- 
sponding to  intermediate  values  of  x  may  be  plotted 
or  drawn   to   the  same  scale,  and  the  curves  may  be 


Fig.  15. 


cut  out  in  paper,  cardboard,  or  zinc  sheet,  and 
fastened  in  vertical  position  relatively  to  each  other 
as  determined  by  their  respective  values  of  ,r,  as  is 
shown  in  Fig.  15. 


50  tNTRODUCTION   TO 

If  _>'  =  —  lO  in  (35),  that  equation  becomes 

60^ 


X. 

z. 

0 

0.00 

I 

+  1.76 

2 

+  3.16 

3 

+  4.28 

4 

+  5-21 

5 

■\-  6.00 

6 

+  6.67 

7 

+  7-24 

8 

+  7-74 

9 

+  8.18 

10 

+  8.57 

If  we  now  give  to  x  consecutive  values  from  o  to 
-[-  10,  we  shall  obtain  the  values  of  x  table  10. 
and  z  in  Table  10,  which  will  enable 
us  to  construct  the  curve  forming  the 
back  side  of  the  model  in  Fig.  15. 
The  last  values  in  the  Tables  9  and  10 
are  identical,  and  they  correspond  to 
the  corner  of  the  model  to  the  rear 
and  right. 

If  (33)  and  (34)  are  combined  with 
each  other  by  the  elimination  of  z, 
we  have 

dxy  2y 

4x  —y   ~   X  ' 

If  f  =  o,  this  equation  is  satisfied  for  any  value 
of  X.  The  equation  will  then  read  0  =  0.  If  .i-  may 
have  any  value  when  j'  =  o,  then  (33)  and  (34)  with 
the  condition  J  =  o  must  be  true  for  any  value  of  x. 
These  equations  both  reduce  to  the  form  zx  =  o. 
Since  this  must  be  true  for  any  value  of  x,  it  follows 
that  z  must  be  zero  when  j  =  o,  and  that  x  may  then 
have  any  value.  If  we  refer  to  Figs.  14  and  15,  we 
find  that  the  axis  x  lies  in  each  of  these  surfaces. 
The  surfaces  therefore  intersect  each  other  along  that 


GRAPHICAL   ALGEBRA.  5^ 

line,    the  condition   of  this  intersection  being  ;:  =  o, 
y  =  o,  A-  =  any  value. 

The   last  equation   must  also  be  satisfied  when  the 
common  factor  j'  is  stricken  out,  or  when 

6^  2 

AX  —  iy~  x' 

Solving  this  equation  for  x,  we  have 

x  =  %±  1/4  -  y (36) 

This  equation  represents  a  curved  line  on  the  hori- 
zontal reference  plane,  directly  under  another  inter- 
section line  of  the  two  surfaces.  By  inspection  of  the 
last  equation  it  is  evident  that  when  j  =  -|-  I  ^^''^ 
±  term  becomes  zero,  and  in  that  case  x  —  \.  If  y 
be  made  greater  than  +  i»  the  ±  term  becomes 
imaginary.  When  y  =  o,  x  becomes  |  ±  |  =  |  or  o. 
This  intersection  line  therefore  crosses  the  axis  x,  the 
other  intersection  line,  at  the  two  points  x  =  o,  x  =  f , 
y  and  2  being  then  both  zero.  (See  Fig.  16.)  For 
all  negative  values  of  y  there  are  two  real  values  of 
X,  symmetrically  related  to  the  value  --{-  f.  The 
curve  representing  these  values  is  shown  in  Fig.  16. 
The  line  of  symmetry  for  which  x  =  f  is  the  line  a,  b 
in  that  figure.  The  point  where  the  curve  crosses 
this  line  is  the  point  where  the  +  term  in  (36)  is 
zero,  and  where  there  is  but  one  value  of  x.  The 
values  of  x  for  various  values  of/  computed  from  (36) 
are  given  in  Table  1 1. 


52 


INTRODUCTION    TO 


— 

y 

9 

h 

S 

\ 

7 

' 

\ 

6 

\ 

5 

1 

\ 

4 

j 

\ 

3 

/ 

\ 

\ 

.? 

/ 

\ 

/ 

1 

/ 

—  X 

3 

2 

A 

0 

J 

o 

3 

4+x 

/ 

2 

a 

+ 

y 

Fig.  i6. 


GRAPHICAL  ALGEBRA. 


53 


It  will  be  of  interest  to 
examine  the  vertical  sec- 
tions of  the  two  surfaces 
along  the  line  a,  b  of  Fig. 
i6. 

In  order  to  do  this  it  is 
only  necessary  to  make 
;tr  =  f  in  equations  (33)  and 
(34)  and  compute  z  for  each 
surface  for  various  values 
of  7  between  -f- |  and  —  i. 

These  equations  become 

-  =  —  3J'> 


TABLE   I  I. 


y- 

.1". 

+  ^ 

0.667 

0 

0.000     or     4-  1.333 

-    I 

-  0.535      ' 

'      +  1.868 

2 

-  0.897      ' 

+  2.230 

—     3 

-  1.1S9      ' 

'      +  2.522 

-      4 

-  1-442      ' 

'      +  2.775 

-     5 

-  1.666      ' 

+  3.000 

-    6 

-  1.872      ' 

'      +  3-205 

-     7 

—  2.062 

'      +  3-395 

-     8 

—  2.239 

'      +  3-572 

-     9 

-  2  407      ' 

'      +  3-740 

—  10 

—  2.565      ' 

'      +  3-898 

1 

\2y 


(33)' 
(34)' 


that 


8  —  gv 

Making  j'  —  -{-  4j   in   these  two  equations,  we    find 
=  —  i-  in  both.      This  is  the  point  where  the 


TABLE  12. 


curve  crosses  the  line  a,  h.  The  values  for  a  few 
points  along  this  line  are  given 
in  Table  12  adjoining.  The 
pupil  should  extend  these  com- 
putations and  draw  the  two  sec- 
tion lines  on  the  same  sheet. 
Compute  the  values  between 
J  =  -["  10  and  J  =  —  10.  In  a 
similar  way  other  and  parallel 
sections  of  the  two  surfaces  may 
be  computed  by  making  x  =l  2,  \  —  i 
3,  or  4,  or  —  2,   —  3,  —  4,  etc., 


z. 

y- 

(33)' 

(34)' 

+:- 

—  2.00 

—  4.00 

+? 

-  1-33 

-  1-33 

0 

—  0.66 

-  0.44 

0.00 

0.00 

—  I 

-f  3.00 

+  0.71 

54  iNTRODUCTiON    TO 

in  (33)  and  (34),  and  then  giving  y  various  values 
between  -f-  10  and  —  10.  These  sections  will  all 
cross  where  y  =  o,  since  z  for  both  surfaces  will  be 
zero.  They  will  also  always  cross  directly  above  the 
curved  line  of  Fig.  16,  the  two  values  of  s  being  there 
equal.  As  will  be  seen  by  (35),  in  which  (34)  is  solved 
for  z,  when  x  and  y  are  positive,  and  x  =  ly,  the 
value  of  z  will  become  infinite.  If  ^x  >  37,  z  will  be 
negative,  and  if  ^x  <  37,  it  will  be  positive. 

16.  We  may  now  assume  the  equation  of  a  plane, 
and  combine  it  with  (33)  and  (34).  The  three  equa- 
tions are 

zx-^-2y  =  o,      .      .      .      (33) 

4XZ  —  T,yz  -\-  6xy  =  o,      ,      .      .      (34) 

X  —  y  -{-  z  =  10.    .      .      .      (37) 

We  have  already  eliminated  z  in  (33)  and  (34)  and 
found  that  the  intersection  projected  on  the  plane 
y,  X  is  the  curve  in  Fig.  16  and  the  axis  x.  The 
equation  of  the  curve  we  have  found  to  be 


:^  =  i±V^,-y (36) 

We  are  to  find  points  common  to  the  three  surfaces. 
These  are  the  points  where  the  intersection  of  (33) 
and  (34)  pierces  plane  (^y).  Either  (33)  or  (34)  may 
be  combined  with  {T,y)  by  eliminating  z,  the  resulting 
equation  being  combined  with  (36). 

Eliminating  z  in  (33)  and  (s/), 

x' -  {10 -^  y)x  =  2y.      .      .      .      (38) 


GRAPHICAL   ALGEBRA.  55 

For  algebraic  solution  it  is  simplest  to  eliminate  j 
in  the  three  original  equations  and  then  find  x  and  z 
in  the  two  simplest  equations  resulting.  For  graphical 
discussion  it  is  more  instructive  to  solve  the  last  equa- 
tion for  X.     The  result  is 


'^  =  5  +  \y  ±  \  ^V  H-  2  8j  +  lOo.      .     (39) 

This  expression  for  x  is  composed  of  two  parts,  which 
may  be  indicated  by 

'f  1  =  5  +  \y, 


x^=  ±\^f  -\-  28j  +  100. 

The  part  of  x  represented  by  x^  is  represented 
geometrically  by  a  straight  line,  which  crosses  the 
axis  y,  {x\  =0,)  where  j  =  —  10,  and  which  crosses 
the  axis  x,  {y  =  O,)  where  x^  =  5.  This  line  is  the 
line  c,  d  in  Fig.  17.  For  the  total  value  of  x  at  any 
point  we  must  add  to  the  values  of  x  for  this  line  the 
±  term  ;i.,.      This  ±  term  becomes  zero  when 

J''  -f-  28j'  -{-  100  —  o, 

or  when  y  =  —  4.202  or  —  23.798.  For  values  of  jj' 
intermediate  between  these  two  the  ±  term  x^ 
becomes  imaginary.  For  values  algebraically  less 
than  —  23.798  (numerically  greater)  or  greater  than 
—  4.202,  the  2b  term  is  real.  The  values  of  x  for 
assumed  values  o(  y  are  given  in  Table  13  as  computed 
from  (39). 


56 


INTEODUCTION    TO 


TABLE    13. 


y- 

X. 

+  4 



14.550  or  - 

0.550 

+  I 

-- 

II. 178   "   - 

0.178 

0 

-- 

10.000   " 

0.000 

—  I 

-- 

8.772   "   + 

0.222 

—  2 

-- 

7.464   "   + 

0.536 

-  3 

-- 

6.000   "   4- 

1. 000 

-  4 

— 

4.000   "   + 

2.000 

—  4.202 

+  2.S94 

-  23.798 

-  6.899 

-  24 

— 

6.000  or  — 

8.000 

-  25 

— 

5.000   "   — 

10.000 

-  28 

— 

4.000   "   — 

14.000 

-  30 

"~" 

3.676   "   - 

16.325 

These  values  of  j  plotted  with  the  double  values  of 
X  determine  the  two  isolated  branches  of  the  curve 
marked  (33),  (37)  in  Fig.  17.  The  line  c,  d  is  a  line 
of  symmetry  with  respect  to  these  branches.  It 
bisects  any  chord  parallel  to  the  axis  x.  Where  this 
line  cuts  the  two  branches  there  is  but  one  value  of  x. 
The  ■±_  term  is  zero. 

By  eliminating  as  before  described,  the  two  values  of 
X  are  found  to  be  ,r  =  -|-  2.8219  and  -r  =  —  4.4886. 
These  are  the  values  of  x  corresponding  to  the  inter- 
section of  the  two  curves  which  these  equations  repre- 
sent. They  are  both  drawn  in  Fig.  17,  one  of  them 
being  reproduced  from  Fig.  16.  These  values  of  x  in 
(36)  or  (38)  give  y  =.  —  4.2009  and  y  =  —  26.1324. 
The  values  x  =  -j-  2.82  19,  j  =:  —  4.2009  locate  the 


GRAPHICAL   ALGEBRA. 


57 


d 

'V 

\ 
\ 

\ 

30 

\ 

V 

\ 
\ 

1 

28 

t 

%^ 

*v 

\ 
\ 

1 

11. 

26 

-> 

/ 

% 

\ 
\ 

/ 

24 

\ 
\ 

22 

?-;p 

\\ 

20 

{3 

34) 

\l 

18 

16 

I  \ 

1    \ 

u 

\ 

\ 

12 

\ 

\ 

V 

10 

1 

1 
\ 

/ 

\ 

\ 

6    ^. 

/ 

\ 

4 

/  \ 
/    \ 
/     \ 

^ 

\ 

A 

'^. 

■v 

1^ 

12 

10 

S! 

Q 

i 

A 

y 

n 

k 

\ 

A 

6        IvV 

10 

■X 

(3 

3)  (I 

U) 

u 

\ 
1 

33  )( 

'34) 

"^ 

/ 

\ 

0 

/ 

+,v 

Fig.  17. 


58  INTRODUCTION    TO 

point  of  intersection  of  the  two  curves  which  is  marked 
VI  in  Pig.  17.  The  values  x  =  —  4.4886,  j  = 
—  26.1324  locate  the  point  marked  7i. 

If  the  first  set  of  values  be  substituted  in  either  of 
the  three  equations  (33),  (34),  or  (37),  the  value  of  ^ 
is  found  to  be  ^  =  -f  2.9772.  The  point  common  to 
the  three  surfaces  is  therefore  above  the  reference 
plane  x,  y,  a  distance  2.9772.  Placing  the  other  set 
of  values  in  either  of  the  three  equations  we  find 
2  =  —  11.6438.  This  common  point  is  therefore 
below  the  plane  x,  y  by  this  distance.  We  observe 
also  that  both  curves  intersect  at  the  origin,  where 
X  =  o  and  y  =  o.  These  values  of  x  and  y  being 
placed  in  {^^2>)  and  (34)  will  satisfy  those  equations  for 
any  value  of  ^.  The  geometrical  meaning  of  this  is 
that  the  axis  a  lies  in  both  of  these  surfaces.  It  is  a 
line  of  intersection  of  these  surfaces.  Putting  x  =  o 
and  J  =  o  in  (37),  we  find  that  3  must  equal  10,  and 
cannot  have  any  other  value  when  these  conditions 
are  imposed.  Another  point  common  to  the  three 
surfaces  is  therefore/  =  o,  ;r  =  o,  ^  =  10. 

We  may  get  additional  evidence  on  this  point  by 
making  ^  =  -f-  10  in  the  equations  of  the  three  sur- 
faces.     They  then  become 

io,r+2j=o, (33)' 

40X  —  30J  +  6xy  =  0,    .      .      .      (34)' 

^-/  =  o (37)' 


GRAPHICAL   ALGEBRA.  59 

It  will  be  seen  that  if  x  =  o,in  any  one  of  these 
equations,  j  is  also  zero.  This  indicates  that  on  the 
horizontal  plane  where  ,3=10  the  sections  of  the 
three  surfaces  cut  through  the  axis  2,  x  and  y  being 
then  zero.  If  s  be  made  any  other  value  than  10, 
this  will  not  then  hold  for  equation  (37),  but  it  will 
hold  for  the  other  two  equations. 

By  giving  x  various  values  from  -|-  5  to  —  5  in  the 
three  equations  last  written,  the  corresponding  values 
of  J/  may  be  computed,  and  the  curves  or  lines  repre- 
sented by  those  equations  may  be  drawn.  This  is 
done  in  Fig.  18. 

The  paper  represents  the  plane  ^  =  -[~  lO-  The 
axis  ;;  is  a  line  at  right  angles  to  the  paper,  at  the 
intersection  of  the  axes  x  and  y. 

The  three  points  common  to  the  surfaces  which  we 
have  found  are 

;tr  =  O,  J  =  O,  ^  =  -[-   10, 

,r  =  + 2.8219,       J' =  —     4.2009,       ::  =  +    2.9772, 

X  =  —  4.4886,        y  =  —  26.  1324,         ^  =  —   I  1.6438. 

It  is  evident  that  by  making  z  =^  -{-  2.9772  or 
—  11.6438  in  the  three  fundamental  equations,  we 
may  compute  and  draw  the  curves  representing  the 
sections  of  the  surfaces  on  either  of  the  horizontal 
planes  thus  determined,  as  has  just  been  done  for  the 
plane  z  =  10. 


6o 


INTRODUCTION    TO 


Furthermore,  by.  making  y  =  —  4.2009  or 
—  26.1324  in  the  fundamental  equations,  similar  sec- 
tions on  the  vertical  planes  thus  determined  may  be 


-y 

Nq^ 

7/ 

4 

^  "i'i  > ' 

• 

\ 

3 

i 

tifjj 

(-'5^; 

^ 

\ 

0 

N 

S.     ^ 

\ 

—  X 

4 

■J 

'^ 

'  / 

1  \\ 

I 

0 

3 

-\-x 

/ 

2 

/ 

3 

K 

/ 

4 

\(37 

) 

('33/ 

/ 

\(:m) 

^ 

\ 

+ 

y 

Fig.  18. 
drawn.     These  planes  are  at  right  angles  to  the  j/-axis. 
The  sections  of  the  surface  will  thusbe  found  to  inter- 
sect at   the  points  determined   by  the  values  x  and  z 


GRAPHICAL    ALGEBRA.  6 1 

above  given.  By  giving  values  to  x  of  -|-  2.8219  and 
—  4.4886  the  sections  at  right  angles  to  the  ;t:-axis 
through  the  points  common  to  the  three  surfaces  may- 
be examined. 

The  pupil  should  not  be  disturbed  that  he  is 
unable  to  form  a  complete  mental  picture  of  these 
surfaces  and  of  their  intersections.  The  methods 
which  have  been  suggested  will  enable  him  to  do  so  if 
he  cares  to  persist  and  do  the  necessary  computing 
for  the  eight  trihedral  angles  determined  by  the  refer- 
ence planes.  It  should,  however,  be  remembered  that 
such  discussions  are  continued  in  the  college  course, 
and  that  there  will  always  renj^in  many  things  yet  to 
be  learned. 

17.  It  is  easy,  by  means  of  four  equations  having 
four  variable  unknown  quantities,  to  find  values  for 
the  quantities  which  will  satisfy  all  of  the  equations. 
The  geometrical  meaning  of  such  equations  has  taxed 
the  powers  of  the  wise  men  for  many  years.  Much 
has  been  written  about  the  properties  of  space  of  four 
dimensions.  But  it  does  not  appear  that  any  one  has 
been  able  to  form  any  satisfactory  physical  or  geo- 
metrical conception  of  what  space  of  four  dimensions 
might  be.  Here  algebra  and  geometry  appear  at 
present  to  part  company.  And  if  that  kind  of  space 
should  come  to  be  understood,  we  should  still  be  in 
the  dark  concerning  space  of  five  or  six  or  ten  dimen- 
sions. 


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